If you had a native proxy, you could configure traffic from the browser to go through the proxy and you would be able to modify requests as needed. I would definitely encourage you to look at this as it could work well.
Even in Manifest V2, we required a blocking webRequest listener to make a decision synchronously. This means we never officially supported speaking to an external service before deciding how to modify or cancel a request. That said, some developers used the XMLHTTPRequest API which does run synchronously and could be used to make a network request to a third-party service.
Manifest V2 will eventually no longer be available - the timeline for enterprise deprecation is June 2025. That said, for extensions installed by policy you can still use the blocking variant of webRequest in Manifest V3. You cannot however use XMLHTTPRequest which does mean that unfortunately your use case is not easily solved right now.
I hope that all makes sense.